Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.
Exact expanded first-order arithmetic statement
forall r d h q t a. (((r + h * d) + (h * h) * q) * (t + h) + a) = ((r * t + a) + h * (d * t + r)) + (h * h) * (q * (t + h) + d)Constructive proof overview
Generated structural guide
The successor Horner transition has an exact subtraction-free quadratic Taylor remainder.
The unchanged tactic script uses 7 declared prerequisites and contains 68 exact native proof lines.
Alpha v34 checked-use · first admitted v25 · independently kernel and Lean verified; not Stable
Proof neighborhood
Direct dependencies
add_mul Stable theorem; checked-use authorized mul_add Stable theorem; checked-use authorized mul_assoc Stable theorem; checked-use authorized mul_comm Stable theorem; checked-use authorized add_assoc Stable theorem; checked-use authorized add_comm Stable theorem; checked-use authorized TH0006 hensel_add_swap_nestedDirect dependents
Formal native tactic body
Dependencies are introduced as named hypotheses before line 1. Local theorem links identify exact declared prerequisites. This exact body belongs to a complete independently kernel-checked constructive proof bundle and has Alpha checked-use authority; it does not imply Stable membership.
Read the argument
Proof checkpoints
This is a reading aid, not a new proof or a proof-tree certificate. Checkpoint groups are consecutive commands, not inferred branch boundaries. Every step links to the preserved script.
Named ingredients (1)
01Fix variables and assumptionsL1–6
02Calculate and transport equalitiesL7–7
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L7
simp [add_mul, mul_add, mul_assoc, add_assoc]
03Establish hrhL8–9
04Establish hdhL10–19
Establish this local claim before using it. It is not an additional assumption. The following proof commands apply mul comm.
05Calculate and transport equalitiesL20–24
06Use earlier factsL25–25
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L25
apply add_comm
07Calculate and transport equalitiesL26–32
08Use earlier factsL33–33
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L33
apply hensel_add_swap_nested
09Calculate and transport equalitiesL34–38
10Use earlier factsL39–39
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L39
apply hensel_add_swap_nested
11Calculate and transport equalitiesL40–42
12Use earlier factsL43–43
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L43
apply hensel_add_swap_nested
13Calculate and transport equalitiesL44–44
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L44
trans (a + (h * r + (h * (d * t) + (h * (h * d) + (h * (h * (q * t)) + h * (h * (q * h)))))))
14Use earlier factsL45–45
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L45
apply hensel_add_swap_nested
15Calculate and transport equalitiesL46–48
16Use earlier factsL49–49
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L49
apply hensel_add_swap_nested
17Calculate and transport equalitiesL50–56
18Use earlier factsL57–57
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L57
apply hensel_add_swap_nested
19Calculate and transport equalitiesL58–66
20Use earlier factsL67–67
Instantiate or apply named facts and discharge the corresponding proof obligations.
- L67
apply add_comm
21Calculate and transport equalitiesL68–68
Carry out the recorded arithmetic or equality steps; inspect the exact commands for their direction and premises.
- L68
refl
Original exact command ledger · 68 lines
- 0001
intro r - 0002
intro d - 0003
intro h - 0004
intro q - 0005
intro t - 0006
intro a - 0007
simp [add_mul, mul_add, mul_assoc, add_assoc] - 0008
have hrh : r * h = h * r - 0009
apply mul_comm - 0010
have hdh : d * h = h * d - 0011
apply mul_comm - 0012
rewrite hrh - 0013
rewrite hdh - 0014
congr - 0015
refl - 0016
trans (h * r + (h * (d * t) + (h * (h * d) + (h * (h * (q * t)) + (a + h * (h * (q * h))))))) - 0017
congr - 0018
refl - 0019
congr - 0020
refl - 0021
congr - 0022
refl - 0023
congr - 0024
refl - 0025
apply add_comm - 0026
trans (h * r + (h * (d * t) + (h * (h * d) + (a + (h * (h * (q * t)) + h * (h * (q * h))))))) - 0027
congr - 0028
refl - 0029
congr - 0030
refl - 0031
congr - 0032
refl - 0033
apply hensel_add_swap_nested - 0034
trans (h * r + (h * (d * t) + (a + (h * (h * d) + (h * (h * (q * t)) + h * (h * (q * h))))))) - 0035
congr - 0036
refl - 0037
congr - 0038
refl - 0039
apply hensel_add_swap_nested - 0040
trans (h * r + (a + (h * (d * t) + (h * (h * d) + (h * (h * (q * t)) + h * (h * (q * h))))))) - 0041
congr - 0042
refl - 0043
apply hensel_add_swap_nested - 0044
trans (a + (h * r + (h * (d * t) + (h * (h * d) + (h * (h * (q * t)) + h * (h * (q * h))))))) - 0045
apply hensel_add_swap_nested - 0046
trans (a + (h * (d * t) + (h * r + (h * (h * d) + (h * (h * (q * t)) + h * (h * (q * h))))))) - 0047
congr - 0048
refl - 0049
apply hensel_add_swap_nested - 0050
trans (a + (h * (d * t) + (h * r + (h * (h * (q * t)) + (h * (h * d) + h * (h * (q * h))))))) - 0051
congr - 0052
refl - 0053
congr - 0054
refl - 0055
congr - 0056
refl - 0057
apply hensel_add_swap_nested - 0058
trans (a + (h * (d * t) + (h * r + (h * (h * (q * t)) + (h * (h * (q * h)) + h * (h * d)))))) - 0059
congr - 0060
refl - 0061
congr - 0062
refl - 0063
congr - 0064
refl - 0065
congr - 0066
refl - 0067
apply add_comm - 0068
refl
Separate complete second-wave branches: Full G095 proof · Alpha v27.